Here my analysis.
Part one: the template
I would compare your template with a creature temporary or like your template pemanent enhanced with the haste spell.
To enhance a creature with the haste spell you need a wizard/sorcerer level 5.
Roudi said:
Accelerated is a template that is added to any living creature that imbibes lulaz once a day.
An accelerated needs one dose of lulaz every day. That is pretty hard. He always needs to carry some stock of lulaz with him.
Roudi said:
Speed: Increase the creature’s base land speed by 15ft.
hmmm a barbarian only gets 10 ft. An accelerated halfling is now as faster as an normal human. This rate is lower than the rate gained by a haste spell.
Roudi said:
Armor Class: The creature gains an inherent +4 dodge bonus to AC.
Haste 3.5 grants only a +1 bonus to AC. In addition to your +4 dodge bonus your template also gains bonus to AC by the bonus to dex. So an accelerated humanoid has at least AC 10 +4 (dodge bonus) +3 (dex bonus).
Roudi said:
Attacks: The creature may make one extra attack each round at her highest base attack bonus when making a full attack action.
This is a very big advantage. A fighter needs six levels to get an extra attack, but his extra attack is not at his highest attack bonus.
In difference to the haste 3.5 spell your accelerated humanoid also can make an extra spell as an extra attack.
Think about an accelerated halfling rogue 1/wizard 1 with two sneak attacks with the ray of frost spell. He wears no armor but with his high dexterity and some magic spells (mage armor or shield) his AC his pretty high. If he runs out of spells he will attack with his short bow twice per round. Level six characters perhaps can survive this encounter but not level five PCs.
Roudi said:
Too Fast for Gravity (Ex): When running, an accelerated moves so fast that she can avoid sinking on semi-solid surfaces. An accelerated may make a full-round run action over surfaces like water or quicksand as if they were solid surfaces. If the accelerated stops running, this ability stops functioning and the character sinks as normal. The accelerated may also make part of a move action up a vertical surface, as per the Up the Walls psionic feat. This ability does not function in heavy armor or while encumbered.
I do not know of a spell in the SRD which duplicates this effect but this ability is permanent and also duplicates the effect of a feat. Is this a spell-like ability ? can this ability dispelled?
Roudi said:
Expeditious Retreat (Sp): 3/day, as the spell, except the duration is one minute per hit die.
No one can outrun an accelerated humanoid
Roudi said:
Blurred Motion (Su): Whenever an accelerated makes a run or full attack action, she acts as though under the effect of a blur spell (20% miss chance) until her next turn.
A permanent blur spell.
Roudi said:
Live Fast Die Young (Ex): The strain put on the accelerated creature’s body by its addiction to lulaz and its various abilities leads the creature to lead a far shorter lifespan. Accelerated creatures only live for 1d4 years after they begin consuming lulaz. Once that time is up, they die as though they had died from natural causes.
This is not really a big disadvantages for adventurers because most of them do not die because of old age
Roudi said:
Restless Body (Ex): An accelerated only needs 4 hours of rest a night. Accelerated spellcasters only require 4 hours of rest before being able to prepare or cast new spells for the day.
Now a adventure party has a guard for most time of the night
Roudi said:
However, an accelerated’s body will not willingly go to sleep. An accelerated must use alchemical or magical means of putting their bodies to sleep in order to rest at all. Also, accelerated creatures are adversely affected by remaining motionless for more than an hour. For each hour spent in a relatively inert state (aside from sleep), the accelerated suffers 1d4 points of nonlethal damage. The number of hours is cumulative in respect to the nonlethal damage dealt each hour; for example, if Jukk is inactive for two hours, he is dealt 1d4 after the first hour and 2d4 after the second hour. Because of the short durations of most sleep spells, and the fast metabolism which works through sleeping potions in half the time, most accelerated creatures suffer from fatigue and insomnia. Some are known to simply try and keep still for a while, letting their restless body knock itself out. An accelerated creature will only awake once all nonlethal damage has been healed or four hours has passed, whichever occurs first.
Accelerated humanoids with a high saves will have a problem with sleeping (thinking of an accelerated paladin

)
The best friend of an accelerated humanoid is the strong fighter of his adventure band who will knock at request the accelerated out.
Roudi said:
Abilities: Increase from the base creature as follows: Dex +6
+6 to dex means +3 to: AC, ranged combat, reflex save and with weapon finesse to bab.
A humanoid with a high dex aquiring the template can not wear any armor because most armor limit the dex bonus to +6.
An accelerated wizard with a high dex can not polymorph himself into a creature with similar high dex.
Roudi said:
Challenge Rating: +2
Level Adjustment: +4
I not good at guessing CRs and LAs

but the LA and CR seem a bit to low.
One additionbal attack + dex bonus + dodge bonus + bonus feats + fast healing is IMHO more than LA +4.